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Variational Quantum Eigensolvers Applied to Problems in High Energy Physics

R. Ferguson

Published Jan 13, 2021• 0 citations
OpenAlex

Abstract

Variational methods have long been used to study strongly-correlated quantum systems. Despite the successes of classical protocols, these are limited in the study of quantum systems, either due to numerical errors like the sign problem or the massive memory requirement to represent large quantum systems. Variational quantum eigensolvers (VQEs) overcome these problems and as a result have been an exciting area of research for the past decade. VQEs use a hybrid classical-quantum set-up, where a cost function (usually the expectation value of an operator) is produced by the quantum processor and then optimized on a classical computer. VQEs have been applied to fields such as quantum chemistry, condensed matter physics, high-energy physics, and classical optimization problems. In this thesis, we investigate VQEs to study gauge theories, which describe fundamental physical interactions and are hard to simulate classically. We use two different approaches in implementing a VQE to study two classes of problems. In the first half of the thesis, we design a VQE to study the U(1) Higgs gauge theory in one spatial dimension. This model is particularly interesting due to the presence of a topological term, which is a candidate to explain the observed CP violation in the universe. In order to perform a quantum simulation of the U(1) Higgs model, the theory is cast on a lattice. In the limit where the lattice size is large and the lattice spacing is small, the continuum limit can be reached and the results correctly resemble the original theory. However, when the lattice is too small or the spacing too big, results are spoiled by finite size effects. In this thesis, we investigate the effects of a finite lattice and determine the parameters to correctly reproduce a first-order phase transition of the continuous theory. Importantly, to resemble a realistic experiment, we include the statistical and intrinsic quantum noise in our VQE. We design the protocol to be applied on a microwave-photonic platform, which is well-suited due to the bosonic nature of the theory's electrical modes. \nIn the second half of the thesis, we design a new class of VQEs, based on a measurement-based quantum computer (MBQC) instead of the typical circuit-based set-up. Rather than performing gates on an initial state, a MBQC transforms an initial state by entangling it with a number of auxiliary qubits which are subsequently measured, resulting in a modification of the initial state. In this part of the thesis, we compare the standard, circuit-based VQE with our newly-designed measurement-based VQE (MB-VQE) through two meaningful examples. First, we introduce a novel way to construct variational state families, using measurement-based techniques, that is more costly to access with circuit-based VQEs. We apply this technique to find the ground state of a two-dimensional, periodic Z2 matter-free lattice gauge theory. In the second approach, we directly translate \nthe circuit-based VQE to a MB-VQE and apply this to the U(1) Schwinger lattice gauge theory, which is used as a benchmarking problem for VQEs. These examples demonstrate that there are specific problems which are better suited for a MB-VQE.

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Variational methods have long been used to study strongly-correlated quantum systems. Despite the successes of classical protocols, these are limited in the study of quantum systems, either due to numerical errors like the sign problem or the massive memory requirement to represent large quantum systems. Variational quantum eigensolvers (VQEs) overcome these problems and as a result have been an exciting area of research for the past decade. VQEs use a hybrid classical-quantum set-up, where a cost function (usually the expectation value of an operator) is produced by the quantum processor and then optimized on a classical computer. VQEs have been applied to fields such as quantum chemistry, condensed matter physics, high-energy physics, and classical optimization problems. In this thesis, we investigate VQEs to study gauge theories, which describe fundamental physical interactions and are hard to simulate classically. We use two different approaches in implementing a VQE to study two classes of problems. In the first half of the thesis, we design a VQE to study the U(1) Higgs gauge theory in one spatial dimension. This model is particularly interesting due to the presence of a topological term, which is a candidate to explain the observed CP violation in the universe. In order to perform a quantum simulation of the U(1) Higgs model, the theory is cast on a lattice. In the limit where the lattice size is large and the lattice spacing is small, the continuum limit can be reached and the results correctly resemble the original theory. However, when the lattice is too small or the spacing too big, results are spoiled by finite size effects. In this thesis, we investigate the effects of a finite lattice and determine the parameters to correctly reproduce a first-order phase transition of the continuous theory. Importantly, to resemble a realistic experiment, we include the statistical and intrinsic quantum noise in our VQE. We design the protocol to be applied on a microwave-photonic platform, which is well-suited due to the bosonic nature of the theory's electrical modes. \nIn the second half of the thesis, we design a new class of VQEs, based on a measurement-based quantum computer (MBQC) instead of the typical circuit-based set-up. Rather than performing gates on an initial state, a MBQC transforms an initial state by entangling it with a number of auxiliary qubits which are subsequently measured, resulting in a modification of the initial state. In this part of the thesis, we compare the standard, circuit-based VQE with our newly-designed measurement-based VQE (MB-VQE) through two meaningful examples. First, we introduce a novel way to construct variational state families, using measurement-based techniques, that is more costly to access with circuit-based VQEs. We apply this technique to find the ground state of a two-dimensional, periodic Z2 matter-free lattice gauge theory. In the second approach, we directly translate \nthe circuit-based VQE to a MB-VQE and apply this to the U(1) Schwinger lattice gauge theory, which is used as a benchmarking problem for VQEs. These examples demonstrate that there are specific problems which are better suited for a MB-VQE.
Quantum-enhanced computing methods are promising candidates to solve currently intractable problems. We consider here a variational quantum eigensolver (VQE), that delegates costly state preparations and measurements to quantum hardware, while classical optimization techniques guide the quantum hardware to create a desired target state. In this work, we propose a bosonic VQE using superconducting microwave cavities, overcoming the typical restriction of a small Hilbert space when the VQE is qubit based. The considered platform allows for strong nonlinearities between photon modes, which are highly customisable and can be tuned in situ, i.e. during running experiments. Our proposal hence allows for the realization of a wide range of bosonic ansatz states, and is therefore especially useful when simulating models involving degrees of freedom that cannot be simply mapped to qubits, such as gauge theories, that include components which require infinite-dimensional Hilbert spaces. We thus propose to experimentally apply this bosonic VQE to the U(1) Higgs model including a topological term, which in general introduces a sign problem in the model, making it intractable with conventional Monte Carlo methods.
<!--HTML-->Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit-based to measurement-based schemes. For each scheme, we provide an application in high energy physics, namely the Schwinger model and Z_2 lattice gauge theory.
The variational quantum eigensolver (or VQE), first developed by Peruzzo et al. (2014), has received significant attention from the research community in recent years. It uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits facing exact modeling of the exponentially growing electronic wavefunction for these many-electron systems. The VQE may be used to model these complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. One important advantage is that variational algorithms have been shown to present some degree of resilience to the noise in the quantum hardware. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm, but without clear descriptions of how the diverse parts fit together. The potential practical advantages of the algorithm are also widely discussed in the literature, but with varying conclusions. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims at disentangling the relevant literature to provide a comprehensive overview of the progress that has been made on the different parts of the algorithm, and to discuss future areas of research that are fundamental for the VQE to deliver on its promises. All the different components of the algorithm are reviewed in detail. These include the representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process to find ground state energies, the post processing mitigation of quantum errors, and suggested best practices. We identify four main areas of future research: (1) optimal measurement schemes for reduction of circuit repetitions required; (2) large scale parallelization across many quantum computers; (3) ways to overcome the potential appearance of vanishing gradients in the optimization process for large systems, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated in a tractable manner. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
Quantum computing is in its greatest upswing, with so-called noisy-intermediate-scale-quantum devices heralding the computational power to be expected in the near future.While the field is progressing toward quantum advantage, quantum computers already have the potential to tackle classically intractable problems.Here, we consider gauge theories describing fundamental-particle interactions.On the way to their full-fledged quantum simulations, the challenge of limited resources on near-term quantum devices has to be overcome.We propose an experimental quantum simulation scheme to study ground-state properties in two-dimensional quantum electrodynamics (2D QED) using existing quantum technology.Our protocols can be adapted to larger lattices and offer the perspective to connect the lattice simulation to low-energy observable quantities, e.g., the hadron spectrum, in the continuum theory.By including both dynamical matter and a nonminimal gauge-field truncation, we provide the novel opportunity to observe 2D effects on present-day quantum hardware.More specifically, we present two variational-quantumeigensolver-(VQE) based protocols for the study of magnetic field effects and for taking an important first step toward computing the running coupling of QED.For both instances, we include variational quantum circuits for qubit-based hardware.We simulate the proposed VQE experiments classically to calculate the required measurement budget under realistic conditions.While this feasibility analysis is done for trapped ions, our approach can be directly adapted to other platforms.The techniques presented here, combined with advancements in quantum hardware, pave the way for reaching beyond the capabilities of classical simulations.
We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE.Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices.As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices.The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %.The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQEs with a few tens of qubits.While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement.Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.